observe

Observe is a do-notation-based DSL library for discrete Bayesian decision theory using the subdistribution monad. The following code solves the student network example from "Probabilistic Machine Learning: Advanced Topics" by Murphy.

student :: Distribution Intelligence
student = do
  diff <- distribution [(Easy, 6/10), (Hard, 4/10)]
  intel <- distribution [(Low, 7/10), (High, 3/10)]
  grade <- case (intel, diff) of
    (Low,  Easy) -> distribution [(A, 30/100), (B, 40/100), (C, 30/100)]
    (Low,  Hard) -> distribution [(A,  5/100), (B, 25/100), (C, 70/100)]
    (High, Easy) -> distribution [(A, 90/100), (B,  8/100), (C,  2/100)]
    (High, Hard) -> distribution [(A, 50/100), (B, 30/100), (C, 20/100)]
  sat <- case intel of
    Low  -> distribution [(Good,  5/100), (Bad, 95/100)]
    High -> distribution [(Good, 80/100), (Bad, 20/100)]
  letter <- case grade of
    A -> distribution [(Positive, 10/100), (Negative, 90/100)]
    B -> distribution [(Positive, 40/100), (Negative, 60/100)]
    C -> distribution [(Positive, 99/100), (Negative,  1/100)]
  observe (grade == C)
  observe (sat == Good)
  return intel

The following code solves the Monty Hall problem.

montyHall :: Distribution Door
montyHall = do
  car <- uniform [Left,Middle,Right]
  choice <- uniform [Left,Middle,Right]
  announcement <- host car choice
  observe (choice == Middle)
  observe (announcement == Left)
  return car

Do-notation with interchange is an internal language for copy-discard monoidal categories. Up to statement interchange, it coincides with the hypergraph representation of a string diagram.

Exact interventions

Observe allows exact interventions in the internal language of the magmoid of normalized stochastic kernels. This allows us to write both observations and interventions in a unified language that does not need a normalization primitive.

conditionalSmoking :: Maybe (Distribution HasCancer)
conditionalSmoking = do
    (gene, isSmoker) <- do 
        gene <- prevalence
        isSmoker <- smokes gene
        return (gene, isSmoker)
    intervene (isSmoker == Smoker)
    hasTar <- tar isSmoker
    cancer <- health gene hasTar
    return cancer

causalSmoking :: Maybe (Distribution HasCancer)
causalSmoking = do
    gene <- prevalence
    isSmoker <- smokes gene
    intervene (isSmoker == Smoker)
    hasTar <- tar isSmoker
    cancer <- health gene hasTar
    return cancer

References.

• A Simple Formal Language for Probabilistic Decision Problems. E. Di Lavore, B. Jacobs, M. Román, 2024.
• Evidential Decision Theory via Partial Markov Categories. E. Di Lavore, M. Román, 2023.
• Getting Wiser from Multiple Data: Probabilistic Updating according to Jeffrey and Pearl, B. Jacobs, 2024.
• Partial Markov Categories. E. Di Lavore, M. Román, P. Sobociński, 2025.
• Accepting Normalization via Markov Magmoids. E. Di Lavore, M. Román, M. Széles, 2025.